Bounded above - Knowlpedia

A bounded above set is a subset $A\subseteq\mathbb R$ for which there exists $M\in\mathbb R$ such that $x\le M$ for every $x\in A$ ; such an $M$ is called an upper bound of $A$ .

Examples

$A=(0,1)$ is bounded above (for example, by $M=1$ ).
$A=\mathbb R$ is not bounded above.

Remarks

This notion depends on the order axioms and is the basic hypothesis needed to define the supremum. A set can be bounded above without having a maximum.